Is it possible to enable backdrop using Python when running Blender without GUI?
You can set this property, you just have to know which area is the node editor. Like this:

import bpy

def get_node_editor_area():
    """
    Get the index of the area containing the compositor
    """
    for i in range(len(bpy.context.screen.areas)):
        area = bpy.context.screen.areas[i]
        if area.type == 'NODE_EDITOR':
            return i

area_index = get_node_editor_area()
bpy.context.screen.areas[area_index].spaces[0].show_backdrop = True


If this is for an addon and the user will have the mouse in the node editor area when this function is called, I think you could use bpy.context.space_data.show_backdrop = True instead.